Euler Lagrange and the Calculus of Variations

In summary, you appear to be confused about how to vary the position and velocity of a 1D system. You found some advice on a website concerning how to do this, and you are now fine.
  • #1
Trying2Learn
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TL;DR Summary
I cannot extract the Euler-Lagrange equation with different variation functions
Good Morning all

Yesterday, as I was trying to formulate my confusion properly, I had a series of posts as I circled around the issue.

I can now state it clearly: something is wrong :-) and I am so confused :-(

Here is the issue:

I formulate the Lagrangian for a simple mechanical system (let's go with 1D)

I formulate the variation of the position and the variation of the velocity, use the Gateaux derivative, work it though..

However, as I understand, I MUST vary the position and velocity INDEPENDENTLY!

And that is where the problem begins (for me)

Could someone read the attached, one page and tell me where I go wrong?

(And if someone can tell me how, I can post a Matlab code that actually tries DIFFERENT eta functions and shows that the correct solutoin produces a stationary Action -- and plots the area between KE and PE)
 

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  • #2
It is puzzling to me why you insist on an approach where position and velocity are varied independently

You appear to be referring to configuration space. For motion in the case of 1D spatial space: in terms of configuration space you have the position coordinate and the velocity coordinate.
Hamiltonian mechanics is formulated in terms of motion in configuration space.Lagrangian mechanics, the mechanics that uses the Euler-Lagrange equation, is formulated in terms of motion in spatial space.

To derive the Euler-Lagrange equation the applicable variation is variation of position.
 
  • #3
Cleonis said:
It is puzzling to me why you insist on an approach where position and velocity are varied independently

You appear to be referring to configuration space. For motion in the case of 1D spatial space: in terms of configuration space you have the position coordinate and the velocity coordinate.
Hamiltonian mechanics is formulated in terms of motion in configuration space.Lagrangian mechanics, the mechanics that uses the Euler-Lagrange equation, is formulated in terms of motion in spatial space.

To derive the Euler-Lagrange equation the applicable variation is variation of position.
Ah ha! You are correct!

I also found this, just now

https://physics.stackexchange.com/q...it-make-sense-to-vary-the-position-and-the-ve

OK, so I am fine.

Sorry to have been a bother.

But thank you, everyone
 

FAQ: Euler Lagrange and the Calculus of Variations

What is the Euler Lagrange equation?

The Euler Lagrange equation is a fundamental equation in the calculus of variations that is used to find the extrema (maximum or minimum) of a functional. It is derived from the principle of least action, which states that the actual path taken by a system is the one that minimizes the action (a mathematical quantity that represents the total energy of the system).

How is the Euler Lagrange equation derived?

The Euler Lagrange equation is derived by setting the functional (a mathematical expression involving a variable function) equal to its variation (a small change in the functional due to a small change in the variable function), and then solving for the variable function. This results in a differential equation known as the Euler Lagrange equation.

What is the significance of the Euler Lagrange equation?

The Euler Lagrange equation is significant because it provides a powerful tool for solving problems in physics and engineering that involve finding the path or function that minimizes a certain quantity. It has applications in a wide range of fields, including mechanics, optics, and economics.

What is the relationship between the Euler Lagrange equation and the calculus of variations?

The Euler Lagrange equation is a central concept in the calculus of variations, which is a branch of mathematics that deals with finding the extrema of functionals. The equation is used to solve problems in the calculus of variations by finding the function that minimizes the functional.

Can the Euler Lagrange equation be extended to multiple variables?

Yes, the Euler Lagrange equation can be extended to multiple variables, resulting in a system of partial differential equations known as the Euler Lagrange equations. These equations are used to find the extrema of functionals that depend on multiple variables, such as in problems involving multiple independent variables or multiple functions.

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